Excellent twitch-gaming skills or a crap-ton of quarters are the only ways to deal with these Demonic Spiders.
Gradius III arcade version: The fireballs in the fire stage, which are present in large numbers and split into indestructible fragments when shot, accompanied by ships that seem to home in on you, making them harder to dodge if you're trying not to shoot; it doesn't help when the passage gets narrow and the fireballs can randomly erupt and kill you without warning. And of course, the infamous Cube Attack, where ice cubes fly across the screen and pile up on the left side, and randomly home in on the Vic Viper at high speed, so you pretty much have to have Jedi reflexes to beat this part; many expert players tweak them to stack up as to form a shield from further oncoming cubes.
Of course, you can easily defeat many enemies with the Energy Laser and four Rotate Options as close to your ship as possible. If you keep charging, you can ram enemies to kill them with the energy build-up, effectively making you all but invincible. When you do this, you lose some of the energy you've charged, which means you're less likely to lose it all by auto-firing (which happens when you're fully charged). Of course, if you're swarmed, you'll probably still be killed unless you can maneuver so that you have enough time to charge some energy before touching each enemy. The four Rotate Options can charge energy, too, making this five times as good.
There is the infamous Option Thief in Gradius II and beyond. In many situations, trying to dodge it will get you killed. Better to lose a couple options than to die and lose everything. At the beginning of V's final stage, there's four of them.
The final stage of Gradius Gaiden has four of them appear from the right just before the final boss, so you're guaranteed to lose all of your options. It's a subversion, though, as the final boss follows the tradition set from the first game.
Spheres Of Chaos has Tidddlers. While most common aliens either are all over the place or chasing player, these manage to do both while being rather fast too. This results in many deaths.
Bacteria and Daisies which are left unkilled can replicate so fast that they cover pretty much most of the screen.
In the Raiden series and its spinoffs Viper Phase 1 and Raiden Fighters, not to mention several of the Toaplan games (like Fire Shark) that inspired Raiden in the first place, exist what the fanbase calls "sniper tanks": tanks that come from unexpected places (under a tree, from a garage, from inside a hut) and immediately shoot at you. Their "dead zone" where they don't fire is exactly as big as their sprite. As the Dynamic Difficulty increases, their shots become faster and more prescient.
And if you think you're safe over the ocean where there are fewer Sniper Tanks, we have Sniper Gunboats, which perform the same role.
Star Force (at least the NES incarnation of it) had a strange enemy shaped like a diamond with an eye in the center. It moved slowly, but followed your ship everywhere. Of course you can't shoot backwards, so you have to hit it from below. Doing so would cause it to fragment into indestructible shrapnel which would hit your ship about seven times out of ten (moreso if you WERE NOT at a great distance). Even worse, this enemy would appear in groups of three to five.
The parasites in Alien Swarm look like the facehuggers of Alien fame. They're hard to spot, hard to shoot, they always spawn in groups, and if they latch onto you, they'll sap all your health away. The only way to get them off is to get to some sort of healing, so if you don't have a dedicated Medic on-hand, you're pretty much screwed. Oh, and more than one can jump on you at once.
In the special "Endurance" stage of Star Wars: Rogue Squadron II, one was required to destroy 999 TIE Fighters and then Darth Vader's special fighter. By itself, this was somewhat doable; but the most frustrating thing was when a TIE Fighter would crash into your ship, destroying both of you. While this is an effective strategy, fans figured the game AI didn't do this intentionally, and referred to this TIE Pilot as the dim-witted "Darth Bob."
The most literal and one of the oldest examples were the spiders from the video game Centipede for their tendency to come out of nowhere and erratic patterns.
Certain enemies in the Touhou games will fire massive swarms of bullets when killed, most notably in Perfect Cherry Blossom stage 4 and Subterranean Animism stage 5. This gets so bad, that it's actually a better tactic to simply not shoot during the majority of these stages.
The first Spy Hunter had Switchblades, who are as fast as you and attack with deadly wheel spikes, and the boat segments had Dr. Torpedos, who fire constant streams of well... torpedoes. The NES-only dolled-up sequel, Super Spy Hunter, had the barrel trucks, who dumped loads of Exploding Barrels in your path that left flaming debris, and the Made of Iron bullet-spamming jets in the plane shmup level.
The phoenixes in the penultimate stage of Abadox. Fireballs gather in the middle of the screen and then fly straight up in the shape of a bird. If you shoot any of these fireballs to try and prevent that from happening, the other fireballs will home in on you relentlessly, and trying to shoot them without powerups is like trying to throw a pebble through a doughnut that's being thrown at you.
The aliens in the Metal Slug series. Not only are they considerably tougher than the standard soldiers (who usually go down in one hit), but their shots are either homing to an almost infallible degree or very fast and hard to dodge.
The boulder-tossing Earth Demons in Chapter 6 of Phelios.